What is the integral of (4x^4+3)/(4x^5+15x+2) ?

The integral of (4x^4+3)/(4x^5+15x+2) is 1/5 ln|4x^5+15x+2|+C

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integral of (4x^4+3)/(4x^5+15x+2)

\int \frac{4 x ^{4} + 3}{4 x ^{5} + 15 x + 2} d x

\frac{1}{5} ln 4 x^{5} + 15 x + 2 + C

Solution steps

\int \frac{4 x ^{4} + 3}{4 x ^{5} + 15 x + 2} d x

Apply u-substitution

: \int \frac{1}{5 u} d u

= \int \frac{1}{5 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{5} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{5} ln ∣ u ∣

Substitute back

u = 4 x^{5} + 15 x + 2

= \frac{1}{5} ln 4 x^{5} + 15 x + 2

Add a constant to the solution

= \frac{1}{5} ln 4 x^{5} + 15 x + 2 + C

ln (sin (4 x))

\frac{\partial}{\partial x} (x y z^{2})

\frac{d}{d y} (x y + y x^{2} + y^{2})

\frac{d}{d x} (2190 \cdot e^{- 0.3 x})

x \to \frac{π}{2} + lim (3 e^{t a n (x)})

What is the integral of (4x^4+3)/(4x^5+15x+2) ?
The integral of (4x^4+3)/(4x^5+15x+2) is 1/5 ln|4x^5+15x+2|+C